下面是 armijo线搜索+最速下降法的小程序，matlab用的很不熟，费了不少劲。

函数：

function g=fun_obj(x)

syms a b

f = 1/2*a^2+b^2-a*b-2*a;

a=x(1);b=x(2);

g=eval(f);

求梯度：

function g=fun_grad(x)

syms a b

f = 1/2*a^2+b^2-a*b-2*a;

gradient = jacobian(f,[a,b]);

a = x(1);b = x(2);

g = eval(gradient);

armijo线搜索：

function mk = armijo( xk, rho, sigma, d )

assert( rho > 0 && rho < 1 );

assert( sigma > 0 && sigma < 0.5 );

mk = 0; max_mk = 100;

while mk <= max_mk

    x = xk + rho^mk * d;

    if fun_obj( x ) <=  fun_obj( xk ) + sigma * rho^mk *fun_grad(xk)*d';

        break;

    end

    mk = mk + 1;

end

return;

主程序：

function result = armijograd(x0)

max_iter = 5000;    % max number of iterations

EPS = 1e-6;         % threshold of gradient norm

rho = 0.45; sigma = 0.2;   % Armijo parameters

k = 0; xk = x0; % initialization

while k < max_iter

    k = k + 1;

    dk =  fun_grad( xk );      % gradient vector

    d = -1 * dk;               % search direction

    if norm( dk ) < EPS        %precision

        break;

    end

    mk = armijo( xk, rho, sigma, d);   %armijo line search

    xk = xk + rho^mk * d;              %update

end

    result = xk;

return;

最终结果是：[4,2]';程序正确。